Let's break down the equation you're asking about and verify its correctness.

The given equation is:

$\int_3^2 f(x) \, dx - \int_3^2 g(y) \, dy = \int_3^2 (f(y) - g(y)) \, dy$

1. Left-hand side:

   • You have two separate integrals: $\int_3^2 f(x) \, dx$ and $\int_3^2 g(y) \, dy$.
   • Both of these are definite integrals over the same interval $[3, 2]$ but for different functions, $f(x)$ and $g(y)$. Since $x$ and $y$ are just dummy variables for integration, the use of different letters doesn’t affect the outcome as long as the functions match.

2. Right-hand side:

   • You have a single integral: $\int_3^2 (f(y) - g(y)) \, dy$, which represents the integral of the difference of the two functions, $f(y)$ and $g(y)$, over the same interval $[3, 2]$.

3. Equality check:

   • The left-hand side is the difference of two integrals, and the right-hand side is the integral of the difference.
   • By the linearity property of integrals, we know that: $\int_3^2 f(y) \, dy - \int_3^2 g(y) \, dy = \int_3^2 (f(y) - g(y)) \, dy$
   • Therefore, the equation is correct, given that $f$ and $g$ are continuous (so that the integrals are well-behaved).

Conclusion:

Yes, the equation is correct by the property of linearity of definite integrals.

Do you want more details or have any questions?

5 related questions:

1. What is the linearity property of integrals, and how is it used?
2. How does the substitution of dummy variables in definite integrals work?
3. Can you explain the meaning of definite integrals over a given interval?
4. What happens if the limits of integration are reversed (from 2 to 3)?
5. Are there specific conditions where linearity of integrals might fail?

Tip:

When dealing with integrals, always ensure the continuity and behavior of the functions being integrated over the specified interval for proper evaluation.